Intersection of a line and an ellipse

Line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ $\dots \dots .(1)\text{and ellipse}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\dots \dots (2)$

Solving equations (1) & (2) we get

$(a^2{m}^2+b^2)x^2+2a^2\mid cmx+a^2(c^2-b^2)=0$

If $D>0$ then $y=mx+c$ is a secant $\mathrm{D}=0$ then $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent $\mathrm{D}<0\mathrm{y}=\mathrm{mx}+\mathrm{c}$ does not meet ellipse

Point form

Equation of tangent to the ellipse at point $(x_1,y_1)$

Let the equation of ellipse be $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$

Then equation of tangent in point form is $\frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}+\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}=1$

Parametric form

Equation of tangent at point $(a\cos \theta ,b\sin \theta )$ to the ellipse is $\frac{x\cos \theta }{a}+\frac{y\sin \theta }{b}=1$

Slope form

$y=mx\pm \sqrt{a^2{m}^2+b^2}$ is a tangent to an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and point of contact is $(\pm \frac{a^{2}m}{\sqrt{a^2{m}^2+b^2}},\overline{+}\frac{b^2}{\sqrt{a^2{m}^2+b^2}})$

Number of tangents through a given point $P(h,k)$

$y=mx+\sqrt{a^2{m}^2+b^2}$ is any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

If it passes through $\mathrm{P}(\mathrm{h},\mathrm{k})$ then

$\mathrm{k}=\mathrm{mh}+\sqrt{{\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2}$

$\mathrm{k}-\mathrm{mh}=\sqrt{{\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2}$

$(\mathrm{k}-\mathrm{mh}{)}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2$

${\mathrm{m}}^2({\mathrm{h}}^2-{\mathrm{a}}^2)-2\mathrm{hkm}+({\mathrm{k}}^2-{\mathrm{b}}^2)=0$

It is a quadratic in $m$ and will give two values of $m$ hence there are two tangents.

Examples

1. If the line $3x+4y=\sqrt{7}$ touches the ellipse $3x^2+4y^2=1$, then the point of contact is

(a) $(\frac{1}{\sqrt{7}},\frac{1}{\sqrt{7}})$

(b) $(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}})$

(c) $(\frac{1}{\sqrt{7}},\frac{-1}{\sqrt{7}})$

(d) None of these

2. The number of values of c such that the line $y=4x+c$ touches the curve $\frac{x^2}{4}+y^2=1$ is

(a) 0

(b) 1

(c) 2

(d) infinite.

3. If $\sqrt{3}bx+ay=2ab$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ then the eccentric angle of the point of contact is

(a) $\frac{\pi }{6}$

(b) $\frac{\pi }{4}$

(c) $\frac{\pi }{3}$

(d) $\frac{\pi }{2}$

4. A tangent having slope of $\frac{-4}{3}$ to the ellipse $\frac{x^2}{18}+\frac{y^2}{32}=1$ intersects the major and minor axes at points $\mathrm{A}$ and $\mathrm{B}$ respectively. If $\mathrm{C}$ is the centre of the ellipses, then the area of the triangle $\mathrm{ABC}$ is

(a) 12 sq.u

(b) 24 sq.u

(c) 36 sq.u

(d) 48 sq.u

5. If the tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ make angle $\alpha$ and $\beta$ with the major axis such that $\tan \alpha +\tan \beta =\lambda$, then the locus of their point intersection is

(a) ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2$

(b) ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{b}}^2$

(c) $x^2-a^2=2\lambda xy$

(d) $\lambda (x^2-a^2)=2xy$

Practice questions

1. If $\mathrm{P}(\mathrm{x},\mathrm{y}),{\mathrm{F}}_1(3,0),{\mathrm{F}}_2(-3,0)$ and $16{\mathrm{x}}^2+25{\mathrm{y}}^2=400$, then ${\mathrm{PF}}_1+{\mathrm{PF}}_2$ equals

(a) 8

(b) 6

(c) 10

(d) 12

2. The length of the major axis of the ellipse

$(5x-10{)}^2+(5y+15{)}^2=\frac{(3x-4y+7{)}^2}{4}$ is

(a) 10

(b) $\frac{20}{3}$

(c) $\frac{20}{7}$

(d) 4

3. Angle subtended by common tangents of two ellipses $4(x-4{)}^2+25y^2=100$ and $4(x+1{)}^2+y^2=4$ at origin is

(a) $\frac{\pi }{3}$

(b) $\frac{\pi }{4}$

(c) $\frac{\pi }{6}$

(d) $\frac{\pi }{2}$

4. The distance of a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ from the centre is 2 . Then the eccentric angle of the point is

(a) $\frac{\pi }{4}$

(b) $\frac{3\pi }{4}$

(c) $\frac{5\pi }{6}$

(d) $\frac{\pi }{6}$

5. If the chord through the points whose eccentric angles are $\theta$ and $\varphi$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ passes through a focus, then the value of $\tan \frac{\theta }{2}\tan \frac{\varphi }{2}$ is

(a) $\frac{1}{9}$

(b) $-9$

(c) $\frac{-1}{9}$

(d) $9$

6. In an ellipse the distance between its foci is 6 and its minor axis is 8 , the eccentricity of the ellipse is

(a) $\frac{4}{5}$

(b) $\frac{3}{5}$

(c) $\frac{1}{\sqrt{52}}$

(d) $\frac{1}{2}$

7. The number of values of $C$ such that the straight line $y=4x+c$ touches the curve $\frac{x^2}{4}+y^2=1$, is

(a) $0$

(b) $2$

(c) $1$

(d) $\mathrm{\infty }$

8. The line $3x+5y=15\sqrt{2}$ is a tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$, at a point whose eccentric angle is

(a) $\frac{\pi }{6}$

(b) $\frac{\pi }{4}$

(c) $\frac{\pi }{3}$

(d) $\frac{2\pi }{3}$

9. Tangents are drawn to the ellipse $3x^2+5y^2=32$ and $25x^2+9y^2=450$ passing through the point $(3,5)$. The number of such tangents are

(a) $2$

(b) $3$

(c) $4$

(d) $0$

10. Tangents are drawn to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ at ends of latus rectum. The area of quadrilateral so formed is

(a) $27$

(b) $\frac{27}{2}$

(c) $\frac{27}{4}$

(d) $\frac{27}{55}$

11. An ellipse passes through the point (4,-1) and its axes are along the axes of coordinates. If the line $x+4y-10=0$ is a tangent to it then its equation is

(a) $\frac{{\mathrm{x}}^2}{100}+\frac{{\mathrm{y}}^2}{5}=1$

(b) $\frac{x^2}{8}+\frac{y^2}{5/4}=1$

(c) $\frac{x^2}{20}+\frac{y^2}{5}=1$

(d) None of these

12. Prove that the line $2x+3y=12$ touches the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=2$

13. The tangent at the point $(4\cos \varphi ,\frac{16}{\sqrt{11}}\sin \varphi )$ to the ellipse $16x^2+11y^2=256$ is also a tangent to the circle $x^2+y^2-2x=15$, find the value of $\varphi$.

14. Find the equations of tangents to the ellipse $9x^2+16y^2=144$ which pass through the point $(2,3)$.

15. The angle between pair of tangents drawn to the ellipse $3x^2+2y^2=5$ from the point $(1,2)$ is ${\tan }^1(12/$ $\sqrt{5}$ )

16. Prove that the portion of the tangent to the ellipse intercepted between the curve and the directrix subtends a right angle at the corresponding focus.

Linked Comprehension Type.

17. For all real $\mathrm{p}$, the line $2\mathrm{px}+\mathrm{y}\sqrt{1-{\mathrm{p}}^2}=1$ touches a fixed ellipse whose axes are coordinate axes.

(i). The eccentricity of the ellipse is

(a) $\frac{2}{3}$

(b) $\frac{\sqrt{3}}{2}$

(c) $\frac{1}{\sqrt{3}}$

(d) $\frac{1}{2}$

(ii). The foci of ellipse are

(a) $(0,\pm \sqrt{3})$

(b) $(0,\pm 2/3)$

(c) $(\pm \sqrt{3}/2,0)$

(d) None of these

(iii). The locus of point of intersection of perpendicular tangents is

(a) ${\mathrm{x}}^2+{\mathrm{y}}^2=5/4$

(b) ${\mathrm{x}}^2+{\mathrm{y}}^2=3/2$

(c) ${\mathrm{x}}^2+{\mathrm{y}}^2=2$

(d) None of these

18. ${\mathrm{C}}_1:{\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{r}}^2$ and ${\mathrm{C}}_2=\frac{{\mathrm{x}}^2}{16}+\frac{{\mathrm{y}}^2}{9}=1$ intersect at four distinct points $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$, Their common tangents form a parallelogram ${\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }{\mathrm{C}}^{\prime }{\mathrm{D}}^{\prime }$.

(i). If $\mathrm{ABCD}$ is a square then $\mathrm{r}$ is equal to

(a) $\frac{12}{5}\sqrt{2}$

(b) $\frac{12}{5}$

(c) $\frac{12}{5\sqrt{5}}$

(d) None of these

(a) $\sqrt{20}$

(b) $\sqrt{12}$

(c) $\sqrt{15}$

(d) None of these

(iii). If ${\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }{\mathrm{C}}^{\prime }{\mathrm{D}}^{\prime }$ is a square, then the ratio of area of the circle ${\mathrm{C}}_1$ to the area of the circumcircle of $\mathrm{\Delta }{\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }{\mathrm{C}}^{\prime }$ is

(a) $\frac{9}{16}$

(b) $\frac{3}{4}$

(c) $\frac{1}{2}$

(d) None of these

19. The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is such that it has the least area but contains the circle $(x-1{)}^2+y^2=1$

(i). The eccentricity of the ellipse is

(a) $\frac{\sqrt{2}}{3}$

(b) $\frac{1}{\sqrt{3}}$

(c) $\frac{1}{2}$

(d) None of these

(ii). Equation of auxilliary circle of ellipse is

(a) ${\mathrm{x}}^2+{\mathrm{y}}^4=6.5$

(b) $x^2+y^4=5$

(c) $x^2+y^4=45$

(d) None of these

(iii). Length of latus rectum of the ellipse is

(a) 2 units

(b) 1 unit

(c) 3 units

(d) 2.5 units

20. The equation of the straight lines joining the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ to the foci of the ellipse $\frac{x^2}{24}+\frac{y^2}{49}=1$ forms a parallelogram. Then the area of the figure formed by the foci of these two ellipse.

(a) 15

(b) 30

(c) 20

(d) 18